Perfect Rigor: A Genius and the Mathematical Breakthrough of the Century — Masha Gessen

Perfect Rigor is the best and most fascinating book I’ve read recently, and it is the sort of book I often seek but too rarely find. The story concerns Grigory Perelman, the man who solved the Poincaré Conjecture and whose eccentricities and life history may or may not be related to his mathematical faculty but certainly make for bizarre, enlightening, and entertaining reading.

Perelman was born into Soviet Russia, a place where the professional study and practice of math were frequently under peril. Soviet math survived Stalinism and the horror of the Soviet Union more generally in part from luck and in part from need, but they suffered from being cut off from the rest of the math world. Still, as Gessen writes:

mathematicians as a group slipped by the first rounds of purges because mathematics was too obscure for propaganda. Over the nearly four decades of Stalin’s reign, however, it would turn out that nothing was too obscure from destruction.

Plus, modern wars cannot be fought successfully without mathematicians. Many, many mathematicians. Math has another useful property from the perspective of Communists living in a resource-deprived, poorly organized society: good math can be done even in conditions of relative privation (which may not be true of, say, engineering).

So math in Russia survived Stalin, even while many other fields suffered. There is a fascinating historical counter-narrative in which Russia evades Communism and Germany evades Nazism via World War I not happening, or not happening the way it did. In that alternate world, tens of millions of people live and contribute to the betterment of humanity. Instead of that world, however, we have the world that World War I bequeathed us and the countless people lost to murderous state machines.

Perelman and his direct family at least were not killed. And in the Soviet Union, math continued to be practiced freely, or mostly freely:

In the after-hours lectures and seminars, the mathematical conversation in the Soviet Union was reborn, and the appeal of mathematics to a mind in search of challenge, logic, and consistency once again became evident. “In the post-Stalin Soviet Union it was one of the most natural ways for a freethinking intellectual to seek self-realization,” said Grigory Shabat, a well-known Moscow mathematician. “If I had been free to choose any profession, I would have become a literary critic. But I wanted to work, not spend my life fighting the censors.”

It was good to do math because there was so little else to do. The many pleasures offered by American or Western European work were not available. Creative freedoms were minimal. Math was among the few places a person could be creative.

Some sections Perfect Rigor are just novel and unknown to me, descriptions of a sub-culture that I’d never thought properly about:

Competitive mathematics is more like a sport than most people imagine. It has its coaches, its clubs, its practice sessions, and, of course, its competitions. Natural ability is necessary but entirely insufficient for success: the talented child needs to have the right coach, the right team, the right kind of family support, and, most important, the will to win. At the beginning, it is nearly impossible to tell the difference between future stars and those who will be good but never great.

I wonder how necessary “the will to win” is, especially given how much later in the book Gessen describes the professional world of math in different terms: “The mathematics community in the United States, and even the world, is very small and very peaceful.” Still, leaving that potential issue aside, the analogy to sport is a powerful one, since sports are more familiar to the average person than math.

More details: Gessen writes of herself:

My own first-grade teacher, in a neighborhood on the outskirts of Moscow that looked just like Perelman’s neighborhood on the outskirts of Leningrad, actually made me pretend my reading skills were as poor as the other children’s, enforcing her own vision of conforming to grade level.

Russia’s many afterschool math clubs did non conform to this bizarre, Harrison Bergeron vision. Which may be why Russia could continue to produce prodigious mathematicians even as much of the rest of its society decayed under the cruelties and absurdities of Communist rule. Those cruelties and absurdities are well-known, and they emerge in the way the Soviet Union sought contradictory goals:

The entire Soviet system of secondary education was based on the concept of uniformity: everyone was to be taught the same thing at the same time, using the same textbooks. But the Soviet Union still craved international prestige—in fact, that need became more and more pronounced as the technological rivalries of the second half of the century heated up.

Uniformity and excellence are mutually exclusive. As often happens, when ideology and reality diverge, ideology gives way, as it did to some extent for Perelman’s school. His school

let him avoid confronting the fact that he lived among humans, each with his or her own ideas and thoughts, to say nothing of emotions and desires. Many gifted children realize with a start as they mature that the world of ideas and the world of people compete for their attention and energy.

Perelman, it appears, never had to choose one over the other. He’s spent his life firmly in the world of ideas, rarely dealing with the world of humans. It is hard to say whether the world or humans or ideas is stranger; presented properly, either can seem strange. Perelman’s life seems strange but also pure and beautiful in a way that I would at times like to emulate but cannot, any more than I think he could emulate my life.

Perfect Rigor speculates some about Perelman’s motives and personality, or personalities, but cannot know them certainly. Sergei Rukshin is Perelman’s first serious math coach, and even very early he is happy with one of Perelman’s interests, or lack of interests: “He was never interested in girls,” unlike many of his classmates, who were caught “doing something as undignified and distracting as kissing a girl.” Life is about trade-offs and on some level Perfect Rigor encourages us to consider some of the tradeoffs some high-level mathematicians make (though not all: Feynman, for example, devotes some stories in Surely You’re Joking, Mr. Feynman! to understanding women).

Maybe lack of sexual interest is in part from the demand side as much as the supply side. Perelman himself is, when he is young, “an ugly duckling among ugly ducklings,” though that changes a little when he is older. One wonders about the links, if any, between physical appearance and math (or other intellectual) skill. The vigorous rejection of “surface” matters seems common among high achievers, though I wonder if I’m letting myself be subject to the availability heuristic.

Used copies of Perfect Rigor on Amazon are gloriously cheap. I don’t know how I missed the book when it first appeared in 2009.